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UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN
UNIVERSAL LIFE POLICIES
Nadine Gatzert
Contact (has changed since initial submission):
Chair for Insurance Management
University of Erlangen-Nuremberg
Lange Gasse 20
D-90403 Nuremberg (Germany)
Tel.: +49/911/5302885
Email: [email protected]
Gudrun Hoermann
Contact (has changed since initial submission):
Leopoldstrasse 106
D-80802 Munich (Germany)
Email: [email protected]
August 2009
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UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN
UNIVERSAL LIFE POLICIES
ABSTRACT
Universal life policies are the most popular insurance contract design
in the United States. They have either a level death benefit paying a
fixed face amount, or an increasing death benefit, which additionally
to a fixed benefit pays the available cash value, and both types include
the option to switch from one to the other. In this paper, we are
interested in the fact that––unlike a switch from level to increasing––a
switch from increasing to level death benefit requires neither fees nor
additional evidence of insurability. To assess the impact of the death
benefit switch option, we develop a model framework of increasing
universal life policies embedding the option. Consideration of
heterogeneity in respect of mortality via a stochastic frailty factor
allows an investigation of adverse exercise behavior. In a
comprehensive simulation analysis, we quantify the net present value
of the option from the insurer’s perspective using risk-neutral
valuation under stochastic interest rates assuming empirical exercise
probabilities. Based on our results, we provide policy
recommendations for life insurers.
Keywords: Increasing death benefit; Death benefit switch option;
Heterogeneity in respect of mortality
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1. INTRODUCTION
First introduced in 1979, universal life is now the most important individual life insurance
contract type in the United States. Lifelong universal life policies offer flexibility with respect
to frequency and amount of premium payments and two death benefit options to choose from
(see Cherin and Hutchins (1987, p. 691) and D’Arcy and Lee (1987, p. 453)). The level death
benefit pays a constant specified face amount; the increasing death benefit pays the available
cash value (or policy reserve) in addition to a fixed face value. Either type of contract
typically embeds the option to switch from level to increasing or vice versa (the death benefit
switch option). A switch from level to increasing benefits requires new evidence of
insurability and, possibly, an extra fee since the death benefit immediately increases by the
current amount of cash value at the time the option is exercised. In contrast, when switching
from increasing to level benefits, the death benefit is fixed at the current value. Thus, in the
latter case, the switch does not affect the net amount at risk, i.e., the difference between death
benefit and cash value, at the switch exercise time and so there are usually no special
requirements or fees involved in making this type of switch (see Smith and Hayhoe (2005, p.
2); see also, e.g., www.sagicorcapitallife.com). However, development of the net amount at
risk after the switch depends on premium payment behavior. Thus, there is some question as
to whether insurers should be concerned about death benefit switches under otherwise
unchanged actuarial assumptions.
In the present paper, we examine the death benefit switch option in a pool of increasing
universal life policies with the goal of enhancing understanding of this feature. To accomplish
this, we develop a model framework for increasing universal life contracts with death benefit
switch option that incorporates heterogeneity in respect of mortality, switch probabilities, and
stochastic interest rates. Based on this model, we evaluate the option under different premium
payment assumptions after switching and for various exercise scenarios. By considering
3
results for adverse exercise behavior depending on an insured’s health status, we derive policy
implications and, in particular, analyze whether a requirement of charges or evidence of
insurability would be advisable.
The literature about universal life insurance is mainly concerned with the return on
universal life policies (e.g., Belth, 1982; Cherin and Hutchins, 1987; Chung and Skipper,
1987; D’Arcy and Lee, 1987). Carson (1996) finds determinants for universal life cash values,
and Carson and Forster (2000) examine policy yields of whole and universal life contracts.
Costs of universal and term life insurance are compared in Corbett and Nelson (1992). Carson
(1996), Cherin and Hutchins (1987), and Chung and Skipper (1987) empirically study the
return characteristics of increasing universal life policies. However, to date there have been no
attempts to develop a model of universal life contracts with increasing death benefit, much
less any study of the death benefit switch option. The same is true regarding premium
payment options in universal life policies. Most studies are restricted to the paid-up option
(i.e., stopping premium payments) in participating life insurance contracts (Kling, Russ, and
Schmeiser, 2006; Linnemann, 2003, 2004; Steffensen, 2002). In addition to the paid-up
option, Gatzert and Schmeiser (2007) integrate the resumption option (i.e., resumption of
premium payments after having made the contract paid-up) in their framework for
participating policies.
To the best of our knowledge, increasing universal life policies and the death benefit
switch option have not yet been studied. We provide an actuarial model framework of a
universal life contract with increasing death benefit and incorporate the death benefit switch
option. Since universal life policies are lifelong contracts that pay a death benefit, we account
for mortality risk as a central risk factor. Mortality varies among insureds, and thus
heterogeneity in respect to mortality is modeled by a stochastic frailty factor on a given
deterministic mortality table. The concept of frailty was originally defined by Vaupel et al.
(1979) in terms of the continuous force of mortality. In this paper, we use the term "frailty
4
factor" in the discrete context in order to express the factor’s stochasticity, as well as
respective distributional characteristics. An examination of adverse exercise behavior with
respect to an insured’s health status is of importance, as exercise of the death benefit switch
option does not require evidence of insurability. The new level death benefit contract is thus
based on unchanged actuarial assumptions.
After switching, premium payments are adjusted since the former increasing policy
premium is no longer adequate for the new level policy. Due to the full flexibility in premium
payments for universal life contracts, the modification is not prescribed by the insurer; the
only restriction is prevention of policy lapse.1 An evaluation of the switch option thus
necessarily involves assumptions about modified premium payment behavior after switch. It
is this combination of options—the death benefit switch option and premium payment
options—that can have substantial negative effects for the insurer. We consider two viable
premium payment scenarios, one with constant premiums and one with flexible payments.
To gain detailed insight into the death benefit switch option of increasing universal life
policies, we conduct a comprehensive investigation for different switch probabilities. In a
simulation analysis, we quantify the net present value of the option using risk-neutral
valuation under stochastic interest rates based on the Vasicek model. We then study the effect
of adverse exercise behavior by assuming different switch probabilities depending on an
insured’s health status and on the time since policy inception (and thus on the amount of
policy cash value). This procedure allows an investigation of the necessity of requiring
evidence of insurability. Finally, we conduct a sensitivity analysis with respect to the frailty
factor distribution.
1 A universal life policy lapses if the cash value is insufficient to pay policy costs (see Carson, 1996, p. 675).
In this case, the contract is terminated without payout to the policyholder. During a one-month grace period,
catch-up premium payments can be made to avoid policy lapse. After that period, reinstatement of the policy
requires new evidence of insurability as well as payment of all outstanding premiums (see Trieschmann, Hoyt, and Sommer, 2005, p. 341). This understanding of policy lapse is in contrast to exercise of the
surrender option, when the cash surrender value of the policy is paid out.
5
Results show that the value of the death benefit switch option is strongly dependent on
premium payment behavior after exercise and on the health status of an exercising insured.
From our findings, we derive policy implications and provide recommendations for insurers,
which can be applied depending on specific––mortality and behavioral––experience in an
insurance portfolio.
The remainder of the paper is structured as follows. Section 2 presents the model
framework, including the model of a universal life policy with increasing death benefit and
the model of the death benefit switch option. In Section 3, the valuation approach is presented
and Section 4 contains numerical results. Policy implications for insurers are discussed in
Section 5; a summary is found in Section 6.
2. THE MODEL FRAMEWORK
The universal life contract with increasing death benefit
We consider a lifelong universal life insurance contract with increasing death benefit. The
policy is issued at time 0t = for an insured of age min, ,x x ω∈ … at inception, where minx is
the minimum entry age admitted. The contract matures at time 1T xω= − + , where ω is the
limiting age of a mortality table, i.e., the one-year probability of dying at age ω , qω′ , is equal
to 1. In what follows, death or survival probabilities based on the mortality table will be
denoted with a prime (`) mark. The one-year table probability of death at age x t+ is thus
given by , 0, , 1x tq t T+′ = −… .
In case of death during policy year t (between time t – 1 and t), the death benefit is paid in
arrears at the end of the year, i.e., at time 1, ,t T∈ … . The increasing death benefit consists
of the sum of a fixed face value Y and the cash value tV at time t :
, 1, ,t tY Y V t T= + = … .
6
To focus on the pure effect of the death benefit switch option in increasing universal life
policies, our model framework does not account for charges or surrenders. According to a
standard actuarial valuation (see, e.g., Bowers et al. (1997) and Linnemann (2004)), for
annual premium payments , 0, , 1tB t T= −… paid at the beginning of each year t in which
the insured is alive, the cash value is given by the following recursive formula:
( ) ( ) ( )1 1 1 11 1 , 1, ,x t t t t x t tq V V B i q Y t T+ − − − + −′ ′− = + ⋅ + − = … , (1)
where 0 0V = . We assume that a constant annual interest rate i is credited to cash value and
premium. Each policy year, this amount is reduced by the cost of insurance, i.e., the product
of death benefit and table probability of death. Calculations are hence based on the actuarial
assumptions of a constant annual interest rate i and probabilities of death according to the
mortality table. With t tY Y V= + , the recursion formula for policy reserves in Equation (1)
reduces to
( ) ( )1 1 11 , 1, , .t t t x tV V B i q Y t T− − + −′= + ⋅ + − = … (2)
Defining the savings premium at time 1t − as ( ) ( ) 1
1 11St t tB V i V
−− −= + − and the cost of insurance
(risk premium) at the same time as ( ) ( ) 1
1 1 1Rt x tB q Y i
−− + −′= + , it turns out from Equation (2) that
( ) ( )1 1 1
S Rt t tB B B− − −= + . From the definition of the savings premium, we obtain the following
expression for the cash value:
( ) ( )1
0
1t
t hSt h
h
V B i−
−
=
= +∑.
Given that ( ) ( )1 1 1
S Rt t tB B B− − −= − , we can also write
( )( )( ) ( )( )( )
( ) ( )
1 11
0 0
1 11
0 0
1 1 1
1 1 .
t tt h t hR
t h h h x hh h
t tt h t h
h x hh h
V B B i B q Y i i
B i Y q i
− −− − −
+= =
− −− − −
+= =
′= − + = − + +
′= + − +
∑ ∑
∑ ∑ (3)
7
Since universal life contracts allow for flexible premium payments, we need to make certain
assumptions in this regard. Universal life policies are usually paid by means of constant
periodic premiums. These constant payments aim to reflect the general savings pattern of life
insurance policies, where savings are accumulated during the earlier years of the contract term
when the costs of insurance are low in order to finance the higher costs of insurance later in
life. We, therefore, base our analysis on constant annual premium payments
, 0, , 1tB B t T= = −… . Given the premium B, the cash value should be positive until
maturity to avoid policy lapse (see Carson (1996, p. 675)). The minimum (constant annual)
premium to fulfill this condition is the amount for which the cash value at maturity equals 0.
Thus, we solve 0TV = for B (see Equation (3)), which is equal to solving the equivalence
principle, and obtain
( )
( )
11
01
0
1
1
Tt h
x hh
Tt h
h
q iB Y
i
−− −
+=
−−
=
′ += ⋅
+
∑
∑. (4)
In general, the net amount at risk tR for a universal life policy at time 1, ,t T∈ … is
given as the difference between the death benefit tY and the cash value tV :
, 1, , .t t tR Y V t T= − = … (5)
In the case of an increasing death benefit, the death benefit at time t is the sum of the
fixed face value and the current cash value, and, thus, the net amount at risk for an increasing
policy is constant and equals the face amount Y throughout the contract term.
Based on the above assumptions, Figure 1 illustrates the premiums, cash value, death
benefit, and net amount at risk of a universal life policy with increasing death benefit from
inception to maturity.
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Figure 1: Premiums, cash value, death benefit, and net amount at risk of universal life policy with increasing death benefit
For constant annual premium payments throughout the policy term—calculated according
to Equation (4)—the cash value first increases and then decreases over time until it becomes
zero at maturity. The decrease in cash value near maturity is due to high costs of insurance at
higher ages that exceed the interest earnings of the cash value and premiums (see Equation
(2)). The death benefit is given by the sum of the fixed face value Y and the cash value and
thus develops analogously to the latter. Hence, the term ”increasing death benefit” is
employed irrespective of the fact that the death benefit may also decrease if the cash value
does. Chung and Skipper (1987) account for this point and use the more precise term “non-
level death benefit.” In insurance practice, however, the term “increasing” is common. It
suggests that the policy––in contrast to a policy with a level death benefit––includes a
dynamic component that increases death benefit coverage in the course of accumulating cash
value. Since the cash value must be positive to keep the policy in force, the increasing death
benefit is always at least as high as a constant level death benefit for the same face amount.
The net amount at risk is equal to the fixed face value Y from policy inception to maturity.
The death benefit switch option
Increasing universal life policies typically give the policyholder the right to switch the
death benefit from increasing to level without charges or additional evidence of insurability.
When exercising the death benefit switch option at time 1, , 1Tτ ∈ −… , the death benefit is
switched to level and fixed at the current value Y Y Vτ τ= + . In our model, the option may be
exercised only once and at discrete exercise times, namely, at the beginning of each policy
0
1
Pre
miu
m
Time0
1
Cas
h Va
lue
Time0
1
Dea
th
Ben
efit
Time
Y
0
1
Net
Am
tat
Ris
k
Time
Y
9
year. Exercise of the option at time τ can also be interpreted as terminating the increasing
death benefit contract and, based on otherwise unchanged actuarial assumptions, purchasing a
new contract with level death benefit Yτ . The death benefit at time t given exercise at time τ
is denoted by
( ) , 1, ,
, 1, ,t
t
Y tY
Y t Tτ
τ
ττ=
= = +
…
…. (6)
When the death benefit switch option is exercised in the accumulation phase of the cash
value, the switch halts further increase of the death benefit by fixing it at the attained level Yτ .
Compared to the case without switch, future death benefit amounts are thus lower until the
increasing death benefit falls below the fixed level again. A switch at or after the peak of the
death benefit curve implies a higher level death benefit until maturity than under increasing
policy conditions. However, in both cases, at the point in time when the switch option is
exercised (and only at this point), the net amount at risk remains unchanged. This is in
contrast to a switch from level to increasing, which immediately increases the death benefit,
and thus the net amount at risk, by the current amount of cash value. Therefore, a switch from
increasing to level does not require any charges or evidence of insurability.
However, future development of net amount at risk depends on future premiums. Hence,
when evaluating the death benefit switch option, it is crucial to take into account possible
changes in premium payment behavior after exercise of the option. When switching before
the peak of the cash value curve, previously calculated premiums for the increasing death
benefit contract (see Equation (4)) are too high for the new level policy. A switch near (some
policy years before), at, or after the peak results in higher premiums due to fixing a higher
death benefit than in the “increasing” case. Thus, it is not possible to simply analyze the death
benefit switch option alone: we need to make assumptions about the premium payment
behavior after switch, which leads to a combined examination of the death benefit switch
option and premium payment options. Again, with universal life policies, policyholders are
10
free to choose the frequency and amount of premium payments as long as the cash value stays
positive.
In the following, we restrict our analysis to two premium payment scenarios that can be
regarded as general cases from the insurer’s perspective as they constitute minimum premium
payment schedules where premiums are just high enough to avoid policy lapse. In particular,
they represent the minimum constant annual premium payments and minimum flexible annual
premium payments that will ensure a positive cash value throughout the contract term. Any
other constant annual or flexible payments keeping the contract in force until maturity need to
exceed these premium amounts.
In the first, “level premium,” scenario, constant annual level premiums ( )B τ paid after the
switch are calculated based on the equivalence principle, taking into account the present cash
value Vτ at the exercise date as an additional single payment. This can be interpreted as
terminating the former increasing death benefit contract and starting a new level death benefit
contract with an initial premium payment in the amount of the current cash value. For
universal life policies, insurers chiefly use constant annual “level” premiums to project policy
values (cash value, cash surrender value, death benefit) that imply a zero cash value at
maturity (so-called policy illustrations). After option exercise, updated policy illustrations are
usually provided. Annual premium notices are often based on the premium values contained
in these projections. Although holders of universal life policies are not forced to pay the stated
premium amount, they likely do so, unless a certain event makes them depart from the
prescribed premium schedule. Since a switch from an increasing to a level death benefit does
not require additional evidence of insurability, mortality and interest rate assumptions remain
the same. The equivalence principle requires the present value of future premium payments to
equal the present value of future benefits (see, e.g., Bowers et al. (1997) and Linnemann
(2004)) i.e.,
11
( ) ( ) ( )1 1
( 1)
0 0
1 1 .T T
t t
t x t x x tt t
B p i V Y p q iτ τ
ττ τ τ τ τ
− − − −− − +
+ + + += =
′ ′ ′+ + = +∑ ∑
If the initial single premium Vτ exceeds the present value of future benefits of the new level
policy, the annual premium is set to zero. Solving for ( )B τ thus yields
( )( )
( )
1( 1)
01
0
1max ,0
1
Tt
t x x tt
Tt
t xt
Y p q i VB
p i
τ
τ τ τ ττ
τ
τ
− −− +
+ + +=
− −−
+=
′ ′ + − = ′ +
∑
∑.
For simplification purposes, we do not include the scenario where, if the available cash value
exceeds the present value of future benefits, the death benefit amount of a universal life policy
might as well be increased in order to maintain a fair contract according to the employed
technical basis. However, this assumption would be favorable from the policyholder’s
perspective and would increase negative effects of the switch option value for the insurer, thus
implying that the obtained switch option value in the present analysis represents a lower
bound to the ‘actual’ option value (which can already be substantial). As regards the
policyholder perspective, we assume that the decision to switch may sometimes be made for
other than financially rational reasons, and that, despite disadvantages in the premium
amount, doing so can still be beneficial for the policyholder, despite the fixed death benefit.
Premium payments at time t for a policy switched at time τ are denoted by
( )( )
, 0, , 1
, , 1,t
B tB
t TB
ττ
ττ
= −= = −
…
…. (7)
The new death benefit ( )tY τ
and new premium payments ( )tB τ
must be taken into account
when calculating cash value ( )tV τ
and net amount at risk ( )tRτ
after exercise of the switch
option, analogously to Equation (1) and Equation (5), respectively.
Figure 2 shows premium payments, cash value, death benefit, and net amount at risk based
on the “level premium” scenario. In Part a), the switch occurs before the peak of the cash
12
value curve; in Part b), the switch occurs at this peak.
Figure 2: “Level premium” scenario––premiums, cash value, death benefit, and net amount at
risk of universal life policy with increasing death benefit switched to level at time τ a) Switch before peak of cash value curve
b) Switch at peak of cash value curve
After the switch, the contract, in principle, works like a traditional whole life insurance
contract with constant premiums. If the switch occurs at time τ before the cash value curve
peaks (see Figure 2, Part a)), premiums drop to constant annual level premiums. These
reduced payments result in slower growth of the cash value. A switch at the peak of the cash
value curve (see Figure 2, Part b)) implies that higher level premiums are necessary, with a
consequent increase of the cash value. The increasing death benefit is fixed at the switch
exercise time. As the cash value increases after switch, the net amount at risk lies below the
constant amount Y in the nonswitch case.
In the second premium payment scenario (“risk premium”), premium payments are
stopped immediately after switch at time τ and not resumed until the cash value is exhausted.
When switching from an increasing to a level death benefit, the death benefit amount is frozen
at the switch exercise time, offering the policyholder the opportunity to maintain the attained
death benefit level by deferring premium payments until depletion of the cash value. From
then on, the risk premium is paid in only such an amount that the cash value remains zero
0
1
Pre
miu
m
Timeτ0
1
Cas
h Va
lue
Timeτ0
1
Dea
th
Ben
efit
Time
Y
τ0
1
Net
Am
tat
Ris
k
Time
Y
τ
0
1
Pre
miu
m
Timeτ0
1
Cas
h Va
lue
Timeτ0
1
Dea
th
Ben
efit
Time
Y
τ0
1
Net
Am
tat
Ris
k
Time
Y
τ
13
until maturity. The premium arrangement is thus based on natural premiums, as they are
related to the amount of benefit. We refer to this setting as a “risk premium scenario” to
emphasize that once the cash value is exhausted, the policyholder must pay the full risk
premium to keep the contract in force. This scenario is typically exercised in the secondary
market for life insurance, where the policies of insureds with reduced life expectancy are
traded. Life settlement companies aim to “optimize” premium payments in the sense of the
risk premium scenario by paying only the minimum premium necessary to keep a policy in
force, speculating on early deaths of the insureds in their portfolio (see, e.g.,
www.settlementwatch.com; www.lifesettlementguide.org; www.idealsettlements.eu;
www.lifesettlementgrp.com).
The above assumptions imply the following formula for premium payments, which is
derived from the recursive development of the cash value in Equation (1), where ( )1tV τ
+ is set to
zero:
( )( ) ( ) ( ) 1
1
, 0, , 1
max 0, 1 , , , 1tx t t t
B tB
q Y i V t Tτ
τ τ
ττ−
+ +
= −= ′ + − = −
…
…. (8)
If the cash value ( )tV τ
at time t exceeds the discounted risk premium for year t (i.e.,
( ) ( ) 1
1 1x t tq Y iτ −+ +′ + ), no premium payment is necessary. Once ( )
tV τ is less than the required risk
premium for the first time, the remainder of ( )tV τ
is exhausted and the outstanding difference
is covered by the premium payment. After the zero level of ( )tV τ
has been reached, it is
sustained by premiums equaling exactly the amount of the discounted annual cost of
insurance ( ) ( ) 1
1 1x t tq Y iτ −+ +′ + . Again, we illustrate the course of premiums, cash value, death
benefit, and net amount at risk in Figure 3.
14
Figure 3: “Risk premium” scenario––premiums, cash value, death benefit, and net amount at
risk of universal life policy with increasing death benefit switched to level at time τ
a) Switch before peak of cash value curve
b) Switch at peak of cash value curve
3. CONTRACT VALUATION
The previous section makes apparent that the effects of the death benefit switch option
depend on switch exercise time and premium payment behavior after switch. When evaluating
the option, however, mortality as a third (random) component also needs to be considered.
Since the option value is determined by the combination of premium payment method, switch
exercise time, and time of death, we account for adverse option exercise behavior. That is, we
consider mortality heterogeneous insureds whose exercise behavior depends on their health
status or mortality expectation. This enables a comprehensive examination of the option and
an investigation of whether fees or evidence of insurability are recommended.
Option valuation can be conducted in different ways depending on policyholder exercise
behavior. Generally, two approaches can be distinguished. First, under financially rational
exercise, policyholders attempt to identify an optimal exercise strategy that maximizes the
option value. This is implemented by solving an optimal stopping problem. In our setting,
determining an optimal exercise strategy is highly ambitious because of the complex
interaction between mortality and financial factors as well as further options embedded in a
0
1
Pre
miu
m
Timeτ0
1
Cas
h Va
lue
Timeτ0
1
Dea
th
Ben
efit
Time
Y
τ0
1
Net
Am
tat
Ris
k
Time
Y
τ
0
1
Pre
miu
m
Timeτ0
1
Cas
h Va
lue
Timeτ0
1
Dea
th
Ben
efit
Time
Y
τ0
1
Net
Am
tat
Ris
k
Time
Y
τ
15
universal life contract. In particular, the switch exercise decision, inter alia, depends on
decisions regarding frequency and amount of premium payments before and after switch,
lapse option exercise, the insured’s health status, and the interest rate. Therefore, tackling this
problem is an extensive undertaking and requires assumptions regarding many decision
variables.
In addition, even though an ever greater number of policyholders may be taking advantage
of increased transparency in the insurance market and are thus making more rational exercise
decisions, empirically observed exercise behavior can still vary from this assumption. Hence,
the option value under rational exercise is likely to overestimate the value actually generated
in insurance portfolios. From the option value based on an optimal exercise strategy, it is
therefore difficult to derive policy recommendations for insurers.
For these reasons, we focus on the second valuation approach and integrate exercise
probabilities into our model. The investigation is conducted from an insurer’s perspective for
a pool of policyholders who do not necessarily exercise their options in a rational way.
Instead, exercise decisions are exogenously made for financial or other, possibly personal,
reasons. In the current market situation, our model allows an assessment of the risk associated
with the switch option in a portfolio of insureds as well as the derivation of policy
implications, as is done in Section 4 and 5, respectively. Thus, in the following, option value
or net present value of the option refers to the value of the death benefit switch option
calculated using this approach.
As data regarding empirical switch exercise behavior are not available, we conduct our
analysis by studying comprehensive exercise scenarios. An insurer can employ the model
using its own switch exercise experience to determine the impact of the switch option in a
portfolio. However, caution is needed when implementing this approach as using exercise
probability estimates from historical data is not entirely without problems because deviations
between actual and estimated probabilities can represent a risk for the insurer.
16
Heterogeneity in respect of mortality
To examine adverse option exercise behavior and thus the impact of an insured’s health
status, we evaluate the death benefit switch option by taking into account heterogeneity in
respect of mortality. As in Hoermann and Russ (2008), we integrate heterogeneity in respect
of mortality by use of a frailty model (see, e.g., Jones (1998, pp. 80–83), Pitacco (2004, p.
14), and Vaupel, Manton, and Stallard (1979, p. 440)). Since the date of the benefit payment
and thus also the amount of premiums paid into the contract depend on insureds’ mortality,
the value of the option to switch from one death benefit scheme to another will also so
depend. The introduction of a frailty factor and, in particular, its stochasticity will allow a
more detailed analysis of the death benefit switch option with respect to the policyholder’s
individual mortality level and exercise behavior.
The one-year individual probability of death for a person age x is obtained by multiplying
an individual frailty factor d with the probabilities of death xq′ of a deterministic mortality
table:
, 1
1, min 0, , : 1 for 0, , and : 1 for 1
0, otherwise
x x
x x
d q d q
q x x d q x q dωω ω ′ ′⋅ ⋅ < ′= = ∈ ⋅ ≥ ∈ = <
ɶɶ … … .
If the resulting product is greater than or equal to 1 for any ages xɶ , the individual probability
of death is set equal to 1 for the youngest of those ages; for all other ages xɶ , it is set to 0. The
random variable ( )K x describes the remaining curtate lifetime of an individual age x. Its
distribution function k xq at a point 0k ∈ℕ is given by
( ) ( ) ( )( ) ( )1
0
1 1 1k
k x k x x hK xh
F k K x k q p q−
+=
= ≤ = = − = − −∏P ,
where k xp is the individual k-year survival probability of an x-year-old and P denotes the
objective (real-world) probability measure. The distribution of the remaining curtate lifetime
17
thus depends on the individual frailty factor. A person with a frailty factor d less than 1
indicates that the insured is impaired with a reduced expected remaining lifetime.
The parameter d can be interpreted as a realization of a stochastic frailty factor D . The
distribution DF of D specifies the portion of individuals whose mortality is lower or higher
than a certain percentage of table mortality. It is characterized as follows (see, e.g., Ainslie
(2000, p. 44), Butt and Haberman (2002, p. 5), Hougaard (1984, pp. 75, 79), and Pitacco
(2004, p. 15)). We assume a continuous frailty distribution such that it can represent fine
differences between remaining life expectancies. It is only defined for positive values of d
and for 0d = , it equals zero. The distribution is right-skewed, i.e., high values of d —
corresponding to high mortalities—can occur. Its expected value is equal to 1, such that the
deterministic mortality table describes an individual with average life expectancy.
For our analyses, we use a distribution that employs as a suitable choice of parameters for
the characteristics listed above, and that is a common choice for frailty models: a gamma
distribution (see, e.g., Butt and Haberman (2002, pp. 8–9), Hougaard (1984, p. 76), Jones
(1998, p. 82), Olivieri (2006, pp. 29–30), and Pitacco (2004, p. 17), all of which refer to
Vaupel et al. (1979, pp. 441–442)). Vaupel et al. (1979) initially chose the gamma distribution
as it is one of the best-known nonnegative distributions, is convenient to work with, and is
very flexible. Although some advantageous properties of the gamma frailty distribution are
lost when applied to a deterministic mortality table instead of a continuous mortality law, it
remains a reasonable assumption. Since mortality probabilities near zero are unrealistic, the
distribution is shifted by a positive value of γ , resulting in a generalized gamma distribution,
( ), ,α β γΓ . For its probability density function, we employ the following formula:
( ) ( ) ( ) ( ) 1
, ,
1, for , , , 0.
d
f d d e dγ
α βα β γ α γ γ γ α β
α β
−−−Γ = − ≥ ∈ >Γ
ℝ
18
Switch probabilities
Let τ denote the time of switch and let ( )s t be the switch probability that depends on the
time t since policy inception. As there is a prescribed development of the cash value in our
setting, dependence on time t can be interpreted as the switch probability depending on the
amount of cash value in the policy. The distribution of τ at a point in time 0k ∈ℕ is given by
( ) ( ) ( ) ( )( )1
1 1
1hk
h
F k k s h sτν
τ ν−
= =
= ≤ = −∑ ∏P .
Moreover, the switch probability can take different values depending on an insured’s health
status measured by the frailty factor d––a realization of DD F∼ , i.e., ( ),s t d . However, in
the following we omit the index d to simplify the notation.
Short-rate process
For the short-rate process, we follow Briys and de Varenne (1994), Hansen and Miltersen
(2002), and Jørgensen (2006) and use the Vasicek model (Vasicek, 1977), which is a
Gaussian Ornstein-Uhlenbeck process. Under the risk-neutral measure Q , the short-rate
process ( )r t evolves as
( ) ( )( ) ( )dr t r t dt dW tκ θ σ= − + Q ,
where ( )( )W tQ , 0 ≤ t ≤T is a standard Brownian motion on a probability space ( )Ω Q, ,F ,
and (Ft), 0 ≤ t ≤ T is the filtration generated by the Brownian motion. The interest rate
volatility σ is deterministic, the mean reversion level is denoted by θ, and the parameter κ
determines the speed of mean reversion.
( )0,P t denotes the price of a zero-coupon bond at time 0 paying $1 at maturity t, where
( )0r r= . Since the zero-coupon bond price in the Vasicek model has an affine term
structure, the expectation can be represented by
19
( )( ) ( )
2 2 22
2 2 300, exp
2 2 4
11
tr u du
t
t
t r te
P e eκ
κσ σ σθ θ
κ κ κκ
−− −= Ε = − − − − − − ∫ −
ℚ . (9)
Hence, once all input parameters have been defined, the entire term structure can be
determined as a function of the current short rate r.
Net present value of the death benefit switch option
Based on the above mortality assumption and the short-rate process, the net present value
(NPV) of an increasing death benefit policy can be calculated as the expected discounted
premium payments less the expected discounted death benefit, using risk-neutral valuation
given a complete, perfect, and frictionless financial market (see, e.g., Björk (2004)). In the
analysis, we assume independence between short-rate and mortality dynamics. Furthermore,
the market is assumed to be risk-neutral with respect to mortality risk, such that the objective
(real-world) probability measure P coincides with the risk-neutral probability measure ℚ
(see, e.g., Bacinello (2003, p. 468) and Dahl (2004, p. 124)). From the insurer’s perspective,
pooling effects are achieved for a sufficient number of policyholders since, at the portfolio
level, only expected values and thus the mortality distribution in the pool are of relevance in
evaluating the contract. According to our assumptions on the frailty distribution, the expected
value of the frailty factor is equal to 1, implying that, on average, mortality in the pool is
described by the deterministic mortality table. For a policy with increasing death benefit
throughout its term, the net present value conditional on D = d under the risk-neutral measure
Q thus results in
( ) ( )( )
( )( )( )
( ) ( )
( ) ( )
( ) ( )
1
0 0
1
0 0
10
1 1
10 0
1 1
10 0
1 1
0, 0, 1 .
t K x
t t
K xr u du r u du
K xt
T Tr u du r u du
tK x t K x tt t
T T
t x t t x x tt t
NPV d B e Y e
B e Y e
B p P t Y p q P t
+
+
− −
+=
− −− −
+≥ == =
− −
+ += =
∫ ∫= Ε ⋅ − Ε ⋅
∫ ∫= Ε ⋅ ⋅ − Ε ⋅ ⋅
= − +
∑
∑ ∑
∑ ∑
Q Q
Q Q (10)
20
When policyholders have the option to switch the death benefit scheme, the stochastic switch
exercise date τ is included in the net present value calculation. For the net present value of
the policy with switch option for D = d, we hence obtain
( ) ( ) ( )( )
( ) ( )( )
( )1
0 0
1 1
10 0
1 1t tT Tr u du r u du
t tK x t K x tt t
NPV d B e Y eτ τ τ+− −− −
+≥ == =
∫ ∫= Ε − Ε ∑ ∑Q Q . (11)
The death benefit ( )1tY τ
+ is given by Equation (6) and premiums ( )tB τ are given by Equations (7)
and (8) for the “level premium” and “risk premium” scenario, respectively.
Equation (11) contains three sources of randomness, namely, the remaining lifetime K(x),
the time of switch τ, and stochastic interest rates. The equation further illustrates that
( )1 K xτ≤ ≤ , i.e., the option can be exercised only as long as the insured is alive. Since we let
the switch rate depend on an insured`s health status and thus on the frailty factor d , switch
probabilities and probabilities of death are dependent. Again, assuming independence
between the stochastic frailty factor and interest rates, Equation (11) can be rewritten as
( )( )
( ) ( ) ( ) ( )( ) ( ) ( )
1 1
10 0
1 0, 1 0, 1T T
t tK x t K x tt t
NPV d
B P t Y P t
τ
τ τ− −
+≥ == =
= Ε − Ε +∑ ∑Q Q
( )( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( )
1
0 1
1
10 1
1 0,
1 0, 1
T T
t K x tt k
T T
t K x tt k
B k k P t
Y k k P t
τ
τ
τ τ
τ τ
−
≥= =
−
+ == =
= Ε = =
− Ε = = +
∑ ∑
∑ ∑
Q
Q
P
P
( ) ( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
1
0 1
1
10 1
0,
0, 1
T Tk
tt k
T Tk
tt k
B K x t k k P t
Y K x t k k P t
τ τ
τ τ
−
= =
−
+= =
= ≥ = =
− = = = +
∑ ∑
∑ ∑
P P
P P
( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )1 1
10 1 0 1
0, 0, 1T T T T
k kt t
t k t k
B k K x t P t Y k K x t P tτ τ− −
+= = = =
= = ≥ − = = +
∑ ∑ ∑ ∑P P P P
( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )1 11 1
10 1 0 11 1
1 0, 1 0, 1 .k kT T T T
k kt t x t t x x t
t k t kh h
B s k s h p P t Y s k s h p q P t− −− −
+ += = = == =
= − − − +
∑ ∑ ∑ ∑∏ ∏
Thus, the expected value of Equations (10) and (11) is obtained by
21
( )( )ENPV NPV D= Q (12)
and
( ) ( ) ( )( )ENPV NPV Dτ τ= Q , (13)
respectively. In the context of heterogeneity in respect of mortality implied by a gamma
distributed frailty factor, closed-form solutions are generally not feasible for the above net
present values.
To assess the value of the death benefit switch option, we subtract the net present value of
the increasing policy without switch in Equation (12) from Equation (13) and denote the value
by OptNPV . Hence,
( )OptNPV NPV NPVτ= − . (14)
4. NUMERICAL ANALYSIS
This section presents results from a simulation study so as to quantify the impact of the
death benefit switch option. First, we consider the increasing universal life contract. Next, we
integrate the switch option and illustrate effects for deterministic switch exercise times and
times of death. We then derive net present values of the option from the insurer’s perspective
for different switch probabilities depending on the health status of insureds and for some
specific exercise scenarios. In addition, a sensitivity analysis with respect to the
parameterization of the frailty distribution is provided.
Input parameters
We examine a universal life insurance contract with increasing death benefit with a policy
face value of Y = $100,000 for a male insured aged x = 45 years at inception. The actuarial
minimum interest rate is set at i = 3.5%. The minimum guaranteed interest rate for universal
22
life products is usually around 4%. For newer products, it is often 3% (see, e.g.,
www.aegon.com). To be conservative, numerical analyses are based on the U.S. 1980
Commissioners Standard Ordinary (CSO) male ultimate composite mortality table with a
limiting age ω = 99. Composite means that smokers and nonsmokers are not distinguished.
An older mortality table with low limiting age––like the 1980 CSO table––is conservative
regarding death risk in the sense that it tends to overstate probabilities of death. In contrast,
modern life tables account for mortality improvement and usually have a limiting age of 120.
For the generalized gamma distribution of the frailty factor, we employ the parameterization
used in Hoermann and Russ (2008), given by ( )2.0;0.25;0.5 .D ∼ Γ The parameter values
lead to a frailty distribution that fulfils the requirements laid out in Section 3. A shift by
0.5γ = means that individual probabilities of death can be at most half the size of the
mortality table probabilities but not less than that. We later vary distributional assumptions to
examine the sensitivity of switch option values to parameterization changes.
For the stochastic interest rate, we use the input parameters given in Hansen and Miltersen
(2002) with speed of mean reversion κ = 0.30723, mean reversion level θ = 3.7%, interest rate
volatility σ= 0.02258, and r(0) = 3.7%. As is common in the life insurance business, the
interest rate credited to the account value (here, 3.5%) is slightly below the interest earned by
the insurance company in the long term (this difference is larger in European countries, e.g.,
in Germany the minimum guaranteed interest rate is currently 2.25%). Numerical results are
derived using Monte Carlo simulation with 50,000 sample paths (see Glasserman, 2004). In
all simulation runs, we use the same set of random numbers to ensure comparability of results.
Value of the universal life contract with increasing death benefit
The constant annual premium for the increasing policy calculated according to Equation
(4) is given by B = $5,937. The risk-neutral net present value from the insurer’s perspective
23
results in NPV = $2,866, as determined by Equation (12) under consideration of stochastic
interest rates and the stochastic frailty factor. More precisely, in a Monte Carlo simulation,
50,000 frailty factors are generated that imply 50,000 individual mortality distributions. Based
on these probabilities of death, the NPV can be determined. It would be zero when using
1D ≡ , i.e., solely the mortality table, as well as the calculation interest rate i instead of
stochastic interest rates.
Value of the death benefit switch option by switch exercise time and time of death
The option to switch from an increasing to a level death benefit can be exercised only once
during the policy term and if done, must be done at the beginning of a year until the year of
death. After switch, premiums are adjusted. In the following, we evaluate the option and
compare results for the two previously described premium scenarios to identify the effect of
future premium payments on the switch option value. In the “level premium” case, constant
annual premiums are paid after switch, which are calculated based on the equivalence
principle, taking the current cash value at the time of switch as a single premium. In the “risk
premium” case, premium payments are stopped at the exercise date and not resumed until the
cash value is exhausted. From then on, the minimum risk premium is paid that will keep the
cash value at zero and thus avoid policy lapse. The OptNPV of the death benefit switch option
is given by the difference between the ( )NPV τ of the increasing policy with switch option and
the NPV of the contract without switch (see Equation (14)).
To provide a first impression of the impact of the death benefit switch option, we calculate
risk-neutral values for different deterministic times of switch exercise and times of death. For
deterministic switch date τ and date of death K(x), Equation (11) simplifies to
( ) ( ) ( )( )
( )( ) ( )( )1
0
0, 0, 1K x
t K xt
NPV B P t Y P K xτ τ τ+
=
= ⋅ − ⋅ +∑.
Note that in order to examine the effect of the switch option in a portfolio, these deterministic
24
values have to be weighted with respective probabilities of switch and survival. Results are
displayed in Figure 4 for “level premiums” (Part a)) and “risk premiums” (Part b)).
Figure 4: Net present value (NPVOpt) of switch option by time of switch exercise and time of death for 45-year-old insureds
a) “Level premium”
b) “Risk premium”
0
10
20
30
40
50
-$80'000
-$60'000
-$40'000
-$20'000
$0
$20'000
111
2131
4151
Time of deathSwitch exercise time
NP
VO
pt
010
2030
4050
-$40'000
-$20'000
$0
$20'000
$40'000
$60'000
$80'000
$100'000
111
2131
4151
Time of death
Switch exercise time
25
If policyholders pay level premiums after switch as shown in Part a), the switch option
value falls below zero for insureds, deceasing after about 40 policy years (age 85). This is
because insureds with high life expectancy “save”––so to speak––on premiums over time
when choosing to exercise the switch option in combination with level premium payments.
Without switch, in contrast, they are likely to survive until the time when the death benefit
decreases again toward maturity (approaching Y). Hence, they do not benefit from the
“increasing” death benefit feature anyway. Option values are lower the earlier the year of
switch and the later death occurs. In the “level premium” case, negative values are thus
generated by insureds with high life expectancy, especially when exercising the switch option
in early policy years.
In the “risk premium” scenario, option values can also become negative from the insurer’s
perspective. This is the case if death occurs early after switch, such that premiums for the
remaining lifetime are covered by the available cash value and no high risk premiums become
due. In contrast, option values are extremely high if death occurs late and risk premiums are
paid after the cash value is exhausted.
Value of the death benefit switch option by switch probability
To obtain the OptNPV of the death benefit switch option, individual switch probabilities,
as well as individual probabilities of death, need to be taken into account. Results for different
constant switch probabilities between s = 0% and s = 100% are displayed in Figure 5 for risk
and level premium payments.
Figure 5 shows that the two premium payment scenarios have very different outcomes. In
the “level premium” case, the net present value from the insurer’s perspective is negative for
all switch probabilities; however, it remains positive in the “risk premium” scenario. In the
latter case, the OptNPV at most reduces to $115 as the switch probability approaches 100%,
implying early switch. Hence, for risk premiums, high net present values (see Figure 4 Part
26
b)) cause the OptNPV of the switch option to always remain positive in the example even
though the probability of occurrence of such extreme events (e.g., survival until t = 45, i.e.,
age 90) is very low. This implies switch profits for the insurer if switch probabilities are
constant in the portfolio of insureds. However, if insureds terminated contracts (policy lapse)
instead of paying high risk premiums after depletion of the cash value, as discussed in Section
2, the OptNPV turns negative, looks similar to the “level premium” curve.
Figure 5: Net present value ( OptNPV ) of switch option for different switch probabilities for
45-year-old insureds
For switch probabilities higher than or equal to 20%, the “level premium” case leads to
negative net present values to about $-2,000 in the calibration employed. This is due to
considerably negative values for early exercise times, which are weighted more heavily for
high switch probabilities (see Figure 4, Part a)). Thus, depending on the premium payment
method, the switch option can have negative effects on an insurer’s portfolio even if switch
probabilities are assumed to be constant over time, an assumption that we will relax in the
following analysis.
Value of the death benefit switch option by switch probability and health status
Since the value of the death benefit switch option is strongly dependent on an insured’s life
expectancy, as demonstrated in Figure 4, we next examine the effect of adverse exercise
-$3'000
-$2'000
-$1'000
$0
$1'000
$2'000
0 2.50% 5% 7.50% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Switch probability
"risk premium" "level premium"
NP
VO
pt
27
behavior with respect to health status on the option’s risk-neutral value OptNPV . This is done
by calculating the option value for switch probabilities that vary depending on an insured’s
individual mortality. We distinguish persons with a frailty factor greater than or equal to one
(d ≥ 1, average or below-average life expectancy) and persons with a frailty factor d < 1
(above-average life expectancy). Results are displayed in Figure 6 for “level premium” (Part
a)) and “risk premium” (Part b)).
The “level premium” graph in Part a) of Figure 6 reveals strong discrepancies in the
OptNPV if the option exercise behavior depends on an insured’s health status. In this case,
from the insurer’s perspective, risk-neutral values remain positive only if persons with above-
average life expectancy have very low switch probabilities and thus tend to switch––if at all––
late in the contract term. The value of the death benefit switch option becomes negative if
they exercise the option with higher probability. This effect is more pronounced the lower the
switch probabilities are for insureds with below-average life expectancy, with the OptNPV
reaching negative values up to about $-3,500. This is in line with results in Figure 4 Part a),
where negative values are generated for early switch times and late times of death.
In the “risk premium” scenario, shown in Part b) of Figure 6, differences depending on the
health status are less distinct, but still visible. In particular, the “risk premium” scenario
generates negative values for the insurer only if persons with below-average life expectancy
exercise the option and switch probabilities are zero for insureds with above-average life
expectancy. This observation is in line with the reasoning that individuals with impaired
health are likely not to pay high risk premiums after depletion of the cash value due to
expectations of early death. If insureds survive until cash value exhaustion, increasing death
probabilities imply high risk premiums and thus lead to positive net present values from the
insurer’s perspective. Altogether, strong adverse effects can be observed.
28
Figure 6: Net present value ( OptNPV ) of switch option for different switch probabilities
depending on health status for 45-year-old insureds a) “Level premium”
b) “Risk premium”
0%2
.5%
5%7.
5%1
0%2
0%30
%40
%50
%6
0%70
%80
%9
0%1
00%
-$4'000-$3'500-$3'000-$2'500-$2'000
-$1'500
-$1'000
-$500
$0
$500
$1'000
$1'500
$2'000
0%
2.5%5%
7.5
%10
%2
0%30
%40
%5
0%6
0%70
%80
%90
%
100
%
Switch probability if d<1 (above-average life expectancy) Switch probability if d>=1
(average or below-average life expectancy)
NP
VO
pt
0%2.
5% 5%7.
5%10
%20
%30
%
40%
50%
60%
70%
80%
90%
100%
-$1'000
-$500
$0
$500
$1'000
$1'500
$2'000
0%2.
5%5%7.
5%10
%20
%30
%40
%50
%60
%70
%80
%90
%10
0%
Switch probability if d<1 (above-average life expectancy)
Switch probability if d>=1 (average or below-average
life expectancy)
NP
VO
pt
29
Additional exercise scenarios
Based on the previous analyses, certain substantial adverse effects can be seen for specific
exercise scenarios that depend on health status and switch option exercise time. If the switch
option is exercised around the time the cash value reaches its peak, for instance, the level
death benefit for the remaining contract term is higher than in the case of the original
increasing death benefit as set out in Section 2. This is because––without switch––the
“increasing” death benefit would actually decrease in line with the cash value, down to the
fixed level Y at maturity (see Figure 1). Hence, when switching, the new level death benefit is
in fact higher than the original death benefit of the increasing contract at certain times during
the contract term. Such comparably higher death benefit amounts can be obtained without
having to pay additional fees or providing new evidence of insurability. Thus, depending on
the insured’s health status, particular exercise behavior can have a considerable influence on
contract value, which may have serious consequences when considering a pool of insureds.
To further emphasize the potential risk of adverse effects regarding the death benefit switch
option, we study several alternative exercise scenarios (see Table 1).
Table 1: Net present values ( OptNPV ) of switch option for specific exercise scenarios
depending on the health status for 45-year-old insureds
s=100%
at t=41 (peak)
s=10%
t=25 to t=41
s=10% to s=100%*
t=25 to t=41
s=10%
t=5 to t=15
All 1d ≥ 1d < All 1d ≥ 1d < All 1d ≥ 1d < All 1d ≥ 1d <
Level
premium -365 12 -377 -750 194 -945 -1’023 293 -1’315 -1’189 856 -2’045
Risk
premium 1’324 -89 1’413 1’332 -108 1’440 1’576 -123 1’700 758 -204 962
Risk
premium
(lapse)
808 10 799 288 -37 325 229 -42 272 -740 128 -867
Notes: 1d ≥ : insureds with average or below-average life expectancy, 1d < : insureds with
above-average life expectancy, *: linear increase.
30
First, for insureds with above-average life expectancy, the switch option is valuable in
combination with the “level premium” scenario. If exercised around the peak of the cash
value curve, a high death benefit is maintained compared to the decreasing benefit that occurs
without switch. Even though new level premiums are higher than the original premiums in
this case, option exercise may still give rise to negative values for the insurer, which can be
observed in Table 1 (first column, “Level premium”). The scenario “s=100% at t=41 (peak)”
compares results when either all insureds, only insureds with average or below-average
( )1d ≥ , or only insureds with above-average life expectancy ( 1d < ) exercise the switch
option with probability 1 at the peak of the cash value curve (i.e., at age 86).
Second, one would suspect that the switch option is especially valuable for insureds with
below-average life expectancy in combination with the “risk premium” scenario. When
exercised around the peak of the cash value curve at t=41 or age 86, persons with reduced life
expectancy preserve a high death benefit without having the underlying mortality table
adjusted. Furthermore, future premiums can mostly be financed from the available cash value.
For insureds with higher-than-average life expectancy, on the other hand, this exercise pattern
would imply high risk premium payments as the policy approaches maturity and thus switch
profits for the insurer. These expectations are confirmed by the numerical results in Table 1
(“s=100% at t=41 (peak)”, “Risk premium”). However, risk-neutral values are much less
negative in this case than they are for adverse exercise by healthy insureds in the level
premium case.
For risk premium payments, we additionally consider a scenario in which policyholders let
the policy lapse, e.g., due to financial distress, as soon as risk premium amounts exceed 10%
of the new level death benefit (first column, “Risk premium (lapse)” in Table 1). The value of
10% was chosen by intuition; however, further tests revealed that results remain robust with
respect to changes in the percentage parameter. Compared to the risk premium scenario
without lapse (second row of Table 1), such behavior has a considerable negative impact on
31
the net present value from the insurer’s perspective if only insureds with above-average life
expectancy are concerned. In particular, the OptNPV is almost cut in half due to the lack of
high risk premium payments. In contrast, if impaired insureds let the policy lapse when high
risk premiums become due, negative effects are alleviated and the net present value is
increased because soon expected death benefit payments do not have to be made.
We further extend the analysis and consider option exercise prior to cash value peak given
a constant switch probability of 10% from age 70 (t=25) to age 86 (t=41) (second column in
Table 1). Results show that option values are substantially affected. In particular, they are
more negative from the insurer’s perspective in the “level premium” case for insureds with
above-average life expectancy. The latter net present value decreases even more if the switch
probability is linearly raised from s = 10% at age 70 to s = 100% at age 86 (third column in
Table 1). There are two effects responsible for these aggravated results. First, if the time of
cash value peak is not the only possible time to switch (but instead ranges from between age
70 and age 86), option exercise, on average, occurs earlier. From Figure 4 Part a) we know
that the earlier the option is exercised by insureds with long remaining lifetime, the lower are
the option values. And second, observed effects are stronger due to the larger number of
insureds still alive at age 70, compared to at age 86, and thus able to exercise the option.
We now turn to the case where the switch option is exercised after five to fifteen policy
years, i.e., between ages 50 and 60 (fourth column in Table 1), given a constant switch
probability of 10%. A reason for switching early during the term of the policy could, e.g., be
the wish to reduce premium payments. In this scenario, results are even more pronounced
than in the case of exercising around the cash value’s peak. As discussed previously, it is
particularly in the “level premium” scenario that adverse exercise behavior by insureds with
high life expectancy generates negative net present values in an insurance portfolio.
These adverse exercise scenarios assume that insureds are well informed about their
individual mortality, i.e., whether they have an above- or below-average life expectancy. The
32
examples indicate that it is those exercise scenarios that are intuitively rational that pose the
greatest threat to insurers: namely, if insureds with above-average life expectancy switch early
and thus “save” risk premiums by making level payments, and if impaired insureds set out
premium payments after switch, being aware that they will possibly not survive until high risk
premiums have to be paid. In fact, the switch option will be even more valuable if insureds
follow optimal exercise strategies to maximize the option value, a topic that is, however,
beyond the scope of this paper.
Sensitivity analysis with respect to the frailty distribution
The variance of life expectancies in a portfolio of insureds can be an important risk driver
when considering policies with death benefit payments. To assess the impact of the frailty
factor distribution, we compare switch profits for different parameterizations of DF and
different switch probabilities, leaving all other parameters unchanged. Part a) of Figure 7
displays the basic scenario with the gamma frailty distribution ( )2;0.25;0.5D Γ∼ with
variance Var(D) = 0.125 (left hand side in Figure 7) and the respective net present values
NPVOpt (right hand side in Figure 7) as shown in Figure 5. The net present value for an
increasing policy is given by NPV = $2,866. We find that varying the frailty distribution has
very little effect on switch profits in the “risk premium” case. For “level premiums,” however,
effects are much more dramatic.
33
Figure 7: Net present value (NPVOpt) of switch option for different parameterizations of the
frailty distribution for 45-year-old insureds
a) Basic scenario
( )2;0.25;0.5D Γ∼
NPVOpt (NPV = $2,866)
b) “Thinner”
( )4;0.125;0.5D Γ∼
NPVOpt (NPV = $1,581)
c) “Wider”
( )6;0.15;0.1D Γ∼
NPVOpt (NPV = $3,584)
d) “Heavy tailed”
( )1.5;0.5;0.25D Γ∼
NPVOpt (NPV = $8,507)
Notes: PDF = probability density function of frailty factor distribution.
0.00.20.40.60.81.01.21.41.61.8
0.0 0.5 1.0 1.5 2.0 2.5
Frailty factor
PD
F
-$6'000
-$5'000
-$4'000
-$3'000
-$2'000
-$1'000
$0
$1'000
$2'000
$3'000
$4'000
0% 2.50% 5% 7.50% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Switch probability
"risk premium" "level premium"
y
NP
VO
pt
0.00.20.40.60.81.01.21.41.61.8
0.0 0.5 1.0 1.5 2.0 2.5
Frailty factor
PD
F
-$6'000
-$5'000
-$4'000
-$3'000
-$2'000
-$1'000
$0
$1'000
$2'000
$3'000
$4'000
0% 2.50% 5% 7.50% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Switch probability
"risk premium" "level premium"
NP
VO
pt
0.00.20.40.60.81.01.21.41.61.8
0.0 0.5 1.0 1.5 2.0 2.5
Frailty factor
PD
F
-$6'000
-$5'000
-$4'000
-$3'000
-$2'000
-$1'000
$0
$1'000
$2'000
$3'000
$4'000
0% 2.50% 5% 7.50% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Switch probability
"risk premium" "level premium"
NP
VO
pt
0.00.20.40.60.81.01.21.41.61.8
0.0 0.5 1.0 1.5 2.0 2.5
Frailty factor
PD
F
-$6'000
-$5'000
-$4'000
-$3'000
-$2'000
-$1'000
$0
$1'000
$2'000
$3'000
$4'000
0% 2.50% 5% 7.50% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Switch probability
"risk premium" "level premium"
NP
VO
pt
34
Part b) of Figure 7 illustrates that a change to a “thinner” frailty factor distribution results
in a much lower net present value of NPV = $1,581 (compared to $2,866). This is because
individuals’ probabilities of death disperse less from average death probabilities according to
the mortality table (Var(D) = 0.0625), i.e., the effect of heterogeneity in respect of mortality
is reduced. Hence, the NPVOpt for high switch probabilities in the “level premium” case is not
as negative as in the basic scenario and values in the “risk premium” case are closer to zero.
Altogether, we find that the difference between the two premium payment scenarios is less
distinct with the thinner frailty distribution.
For the “wider” gamma distribution shown in Part c) of Figure 7 where Var(D) = 0.135,
switch values for “level premiums” decrease compared to the base case, which is particularly
important for negative results at high switch rates.
Tremendous differences can be observed for the comparatively “heavy tailed” distribution
in Part d) of Figure 7. This assumption implies a greater variance of life expectancies in the
portfolio (Var(D) = 0.375). Changes can also be observed for the NPVOpt in the “risk
premium” case. Net present values are much higher, and a peak around a switch probability of
5% is more pronounced. The “level premium” curve decreases substantially over all switch
probabilities. The NPV of the policy without switch option nearly triples to NPV = $8,507.
Thus, even though the main results are essentially robust, this sensitivity analysis
demonstrates the importance and the impact of heterogeneity in respect of mortality in a
portfolio, as well as the relation between premium payment method and mortality distribution.
5. POLICY IMPLICATIONS FOR AN INSURER
Our results do not have straightforward implications for insurance companies. In particular, it
turns out not to be sufficient to simply require evidence of insurability or impose additional
fees in order to reduce the risk inherent in the death benefit switch option. Instead, we
35
identified four key factors that are of relevance for the option value and that must be
considered simultaneously when taking action: insureds’ life expectancies, the chosen
premium payment method after switch, switch probabilities (and thus the time of switch), and
lapsation. It is the combination of these factors that can make the switch option either
valuable or risky for an insurer.
The first question for an insurer is whether to even offer the switch option. As the demand
for insurance protection can decrease or increase over time, policyholders might choose to
surrender if switching is not included in the contract. Signing a new contract, however, has
several disadvantages: evidence of insurability is required, updated actuarial pricing
assumptions may be applied, and charges have to be paid to initiate the contract. Hence, a
switch may be more attractive than surrendering the policy. From the insurer’s perspective,
offering the option to switch from increasing to level has the advantage of keeping those
contracts in its book of business and of reducing surrender rates. In this case, careful
monitoring of the four factors listed above––including empirical switch probabilities in the
pool of policyholders, possible adverse exercise scenarios, and the mortality distribution in
the portfolio of insureds––is vital to avoid risks in the portfolio that originate from switch
option exercise.
Overall, there are several reasons why the switch option is of practical interest to insurers.
First, the option can become valuable when exercised early as well as late during the contract
term, depending on the respective premium payment scenarios. The latter might even become
more important in the future given demographic development and longevity risk, i.e., if
insureds have longer life expectancies. Second, the option is also relevant in that the
opportunity to switch might prevent some policyholders from surrendering the contract.
Third, our analysis of the NPV of the switch option shows that in a pool of insureds for given
switch probabilities, the switch option can have a substantial value, even though many
insureds in the pool may not survive to higher ages when the value of the option is most
36
intuitive.
Given empirical exercise probabilities and the corresponding premium payment behavior,
our model allows insurers to check whether their portfolios might be negatively affected by
the switch option. For instance, if an insurer observes that, typically, constant level premiums
are paid after switch with an annual switch probability of about 5%, caution is advised as
negative values can result from the insurer’s perspective, given the contract calibration in our
examples (see Figure 5). If policyholders tend to stop premium payments after switch,
implications are not as obvious and must be analyzed in more detail. In particular, adverse
exercise experience may pose a risk for insurers if it is mostly the impaired individuals who
exercise this way.
If monitoring reveals possible negative net present values for an insurer, action should be
taken to reduce the risk by considering the four key factors. First, requiring new evidence of
insurability before allowing policyholders to switch from increasing to level death benefits
could help identify an insured’s health status. This would, in principal, allow the adjustment
of actuarial pricing assumptions and, in particular, the mortality table in the case of impaired
individuals. However, since the requirement of providing evidence of insurability, and its
costs, would apply to all insureds and thus penalize healthy insureds, such a requirement
could have the effect of intensifying adverse effects.
To reduce negative effects originating from adverse exercise behavior of healthy insureds
who pay level premiums after switch, adequate charges for the death benefit switch option
could be imposed. In general, fees should be borne by the group of insureds causing the
undesirable adverse effect. However, as the switch option value is strongly linked to the
premium payment method after switch and to the time of switch, charges can hardly be
calculated independent of these factors. A solution would be the prescription of premium
payments after switch, combined with charges to avoid adverse effects. In our examples,
requiring level premium payments after switching means that healthy insureds are charged
37
higher premiums than impaired individuals. Yet, this approach would also imply a change
from universal life to whole life contracts and thus a loss in flexibility for policyholders.
Furthermore, due to the dependence of the switch option value on the time the option is
exercised (in our examples, negative values were predominantly generated for early exercise
times), insurers could restrict switch exercises to predefined time ranges to control for adverse
effects. Finally, if a shift toward rational exercise behavior is noticed, premium pricing needs
to be adjusted based on the maximum option value determined as the solution of an optimal
stopping problem.
6. SUMMARY
Universal life policies with increasing death benefit as well as the death benefit switch
option have not been investigated in the literature to date. In this paper, we develop an
actuarial model framework and conduct a detailed examination of this option. The model
includes heterogeneity in respect to mortality using a frailty model and switch probabilities.
We point out situations where the death benefit switch option can have considerably negative
effects on an insurer and we provide policy implications to reduce the existing risk potential.
One main finding is that the value of the death benefit switch option is strongly dependent
on premium payment behavior after exercise and on the health status of the exercising
insured. A switch in the “risk premium” scenario has predominantly positive effects in the
examples considered, but the option can actually generate severe negative net present values
from the insurer’s perspective in the “level premium” case. Both scenarios share the result
that option values decrease with increasing switch probability, i.e., the greater the number of
insureds who switch early in the contract term, the more the option values decrease. However,
the extent varies when exercise probabilities differ depending on insureds’ life expectancies.
In the case of risk premium payments, negative values occur if it is only impaired persons
38
who switch early in the contract term, while in the level premium scenario, it is insureds with
good health status who generate highly negative values. Similar results are obtained if policies
are switched at or near the peak of the cash value curve, logging in highest possible death
benefit values. Altogether, we find that combined exercise of the switch option and premium
payment options can generate substantial negative net present values from the insurer’s
perspective due to adverse effects regarding insureds’ health status.
Results are stable with respect to parameterization of the frailty distribution. However, the
spread between positive results in the risk premium scenario and negative results in the level
premium scenario is enhanced with greater variance of life expectancies, i.e., heterogeneity of
insureds’ mortality. Hence, careful consideration and estimation of the mortality distribution
in an insurance portfolio is crucial.
In summary, our findings indicate that the death benefit switch option can pose a threat to
insurers in case of adverse exercise behavior with respect to insureds’ health status. This
result depends on the premium payment method after switching and is even intensified when
additionally considering the amount of cash value as a trigger for option exercise. Overall,
insurers should be aware of the potential impact the death benefit switch option can have and
should consider implementing risk reduction measures. Our policy implications are based on a
broad analysis from an insurer’s perspective for a pool of insureds covering a wide range of
possible exercise scenarios. Depending on the observed exercise behavior in an insurance
portfolio, insurers could require evidence of insurability or charge fees in case of option
exercise, prescribe the premium payment method after exercise, or restrict possible option
exercise times. If insureds followed an optimal exercise strategy, resulting switch option
values could in fact be much higher. Determination of the latter would be an interesting
subject of further research, but also a very challenging one due to complex interactions
between frequency and amount of premium payments before and after switch, lapsation, the
insured’s health status, and interest rates.
39
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